#6366143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365694
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,7)(3,9,11,15)(4,13,14,8)(6,10,16,12)(17,18)(19,24)(20,29)(21,27)(22,31)(23,26)(25,30)(28,32);(1,3)(2,6)(4,10)(5,11)(7,16)(8,15)(9,13)(12,14)(17,19,23,20)(18,24,26,29)(21,30,22,28)(25,31,32,27);(17,20,23,19)(18,25,26,32)(21,30,22,28)(24,31,29,27);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(18,26)(24,29)(25,32)(27,31);(2,7)(3,11)(8,13)(10,12)(17,21,23,22)(18,27,26,31)(19,28,20,30)(24,32,29,25)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339696, #72339709, #72339788
maximal normal subgroup of	#72339696, #72339709, #72339788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273339);